Discretized Path Integral Research Articles

Commuting simplicity and closure constraints for 4D spin-foam models

Spin-foam models are supposed to be discretized path integrals for quantum gravity constructed from the Plebanski–Holst action. The reason for there being several models currently under consideration is that no consensus has been reached for how to implement the simplicity constraints. Indeed, none of these models strictly follows from the original path integral with commuting B fields, rather, by some nonstandard manipulations one always ends up with non-commuting B fields and the simplicity constraints become in fact anomalous which is the source for there being several inequivalent strategies to circumvent the associated problems. In this paper, we construct a new Euclidian spin-foam model which is constructed by standard methods from the Plebanski–Holst path integral with commuting B fields discretized on a 4D simplicial complex. The resulting model differs from the current ones in several aspects, one of them being that the closure constraint needs special care. Only when dropping the closure constraint by hand and only in the large spin limit can the vertex amplitudes of this model be related to those of the FKγ model but even then the face and edge amplitude differ. Interestingly, a non-commutative deformation of the BIJ variables leads from our new model to the Barrett–Crane model in the case of γ = ∞.

An efficient computational method for the implementation of a semi-classical instanton approach using discretized path integrals

In the present paper, a numerical method for a semi-classical instanton method was examined. We implemented the instanton approach using discretized path integrals. The computational accuracy of the method is controlled by the following two parameters: the imaginary time duration (τ) and the time increment (Δτ), which represents the discretized path integral. To obtain accurate results, a long τ must be used in combination with a short Δτ; however, because the computational cost is virtually proportional to τ/Δτ, the instanton calculations were computationally expensive under these conditions. In the present study, we propose a method that reduces the computational cost and represents long τ instanton trajectories by employing an extended instanton trajectory from calculations based on a short τ. We applied the method to calculate tunnel splitting in a HO2 hydrogen transfer reaction using the double many-body extension IV potential as a validation.

Spin-foam fermions: PCT symmetry, Dirac determinant and correlation functions

We discuss fermion coupling in the framework of spin-foam quantum gravity. We analyze the gravity-fermion spin-foam model and its fermion correlation functions. We show that there is a spin-foam analogue of PCT symmetry for the fermion fields on a spin-foam model, which is proved for spin-foam fermion correlation functions. We compute the determinant of the Dirac operator for the fermions, where two presentations of the Dirac determinant are given in terms of diagram expansions. We compute the fermion correlation functions and show that they can be given by Feynman diagrams on the spin-foams, where the Feynman propagators can be represented by a discretized path integral of a world-line action along the edges of the underlying 2-complex.

The Construction of Spin Foam Vertex Amplitudes

Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be seen to be in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second keeps closer contact to the structure of Loop Quantum Gravity and tries to identify an appropriate set of constraints to define a Lorentz-invariant interaction of its quanta of space. This approach is in the tradition of the work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine.

Models for modules: the story of

We recall the structure of the indecomposable sl(2) modules in the Bernstein–Gelfand–Gelfand category . We show that all these modules can arise as quantized phase spaces of physical models. In particular, we demonstrate in a path integral discretization how a redefined action of the sl(2) algebra over the complex numbers can glue finite dimensional and infinite dimensional highest weight representations into indecomposable wholes. Furthermore, we discuss how projective cover representations arise in the tensor product of finite dimensional and Verma modules and give explicit tensor product decomposition rules. The tensor product spaces can be realized in terms of product path integrals. Finally, we discuss relations of our results to brane quantization and cohomological calculations in string theory.

SU(2) COHERENT STATE PATH INTEGRALS LABELED BY A FULL SET OF EULER ANGLES: BASIC FORMULATION

We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.

Perfect discretization of reparametrization invariant path integrals

To obtain a well defined path integral one often employs discretizations. In the case of gravity and reparametrization invariant systems, the latter of which we consider here as a toy example, discretizations generically break diffeomorphism and reparametrization symmetry, respectively. This has severe implications, as these symmetries determine the dynamics of the corresponding system. Indeed we will show that a discretized path integral with reparametrization invariance is necessarily also discretization independent and therefore uniquely determined by the corresponding continuum quantum mechanical propagator. We use this insight to develop an iterative method for constructing such a discretized path integral, akin to a Wilsonian RG flow. This allows us to address the problem of discretization ambiguities and of an anomaly--free path integral measure for such systems. The latter is needed to obtain a path integral, that can act as a projector onto the physical states, satisfying the quantum constraints. We will comment on implications for discrete quantum gravity models, such as spin foams.

State and parameter estimation using Monte Carlo evaluation of path integrals

Transferring information from observations of a dynamical system to estimate the fixed parameters and unobserved states of a system model can be formulated as the evaluation of a discrete time path integral in model state space. The observations serve as a guiding potential working with the dynamical rules of the model to direct system orbits in state space. The path integral representation permits direct numerical evaluation of the conditional mean path through the state space as well as conditional moments about this mean. Using a Monte Carlo method for selecting paths through state space we show how these moments can be evaluated and demonstrate in an interesting model system the explicit influence of the role of transfer of information from the observations. We address the question of how many observations are required to estimate the unobserved state variables, and we examine the assumptions of Gaussianity of the underlying conditional probability.

From lattice BF gauge theory to area–angle Regge calculus

We consider Riemannian 4D BF lattice gauge theory, on a triangulation of spacetime. Introducing the simplicity constraints which turn BF theory into simplicial gravity, some geometric quantities of Regge calculus, areas, and 3D and 4D dihedral angles, are identified. The parallel transport conditions are taken care of to ensure a consistent gluing of simplices. We show that these gluing relations, together with the simplicity constraints, contain the constraints of area–angle Regge calculus in a simple way, via the group structure of the underlying BF gauge theory. This provides a precise road from constrained BF theory to area–angle Regge calculus. Doing so, a framework combining variables of lattice BF theory and Regge calculus is built. The action takes a form à la Regge and includes the contribution of the Immirzi parameter. In the absence of simplicity constraints, the standard spin foam model for BF theory is recovered. Insertions of local observables are investigated, leading to Casimir insertions for areas and reproducing for 3D angles known results obtained through angle operators on spin networks. The present formulation is argued to be suitable for deriving spin foam models from discrete path integrals and to unravel their geometric content.

Simplicity and closure constraints in spin foam models of gravity

We revise the imposition of various constraints in spin foam models of four-dimensional general relativity. We argue that the usual simplicity constraint must be supplemented by a constraint on holonomies and together they must be inserted explicitly into the discretized path integral. At the same time, the closure constraint must be relaxed so that the new constraint expresses covariance of intertwiners assigned to tetrahedra by spin foam quantization. As a result, the spin foam boundary states are shown to be realized in terms of projected spin networks of the covariant loop approach to quantum gravity.

Path-integral virial estimator based on the scaling of fluctuation coordinates: Application to quantum clusters with fourth-order propagators

We first show that a simple scaling of fluctuation coordinates defined in terms of a given reference point gives the conventional virial estimator in discretized path integral, where different choices of the reference point lead to different forms of the estimator (e.g., centroid virial). The merit of this procedure is that it allows a finite-difference evaluation of the virial estimator with respect to temperature, which totally avoids the need of higher-order potential derivatives. We apply this procedure to energy and heat-capacity calculations of the (H(2))(22) and Ne(13) clusters at low temperature using the fourth-order Takahashi-Imada [J. Phys. Soc. Jpn. 53, 3765 (1984)] and Suzuki [Phys. Lett. A 201, 425 (1995)] propagators. This type of calculation requires up to third-order potential derivatives if analytical virial estimators are used, but in practice only first-order derivatives suffice by virtue of the finite-difference scheme above. From the application to quantum clusters, we find that the fourth-order propagators do improve upon the primitive approximation, and that the choice of the reference point plays a vital role in reducing the variance of the virial estimator.

DISCRETIZED BRST TRANSFORMATIONS AND QUANTUM MECHANICAL POTENTIAL OF A FREE PARTICLE ON SD

We evaluated a quantum mechanical potential of a free particle on D-dimensional sphere. This system has the second-class constraints. We change the second-class constraints into first-class ones with new canonical variables. A BRST transformation is induced by the first-class constraints and is discretized. A discretized BRST invariant path integral is considered and the quantum mechanical potential is evaluated as R/12.

Spin foam models of Yang–Mills theory coupled to gravity

We construct a spin foam model of Yang–Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barrett–Crane ansatz. In the Euclidean gravity case, we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidean gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang–Mills theory coupled to gravity.

Understanding hadron structure using lattice QCD

An elementary introduction is presented to the study of hadron structure using lattice QCD. Following a brief review of relevant aspects of path integrals, the discrete lattice path integral is presented for gluon and quark fields and used to calculate physical observables. Essential aspects of instanton physics are reviewed, and it is shown how the instanton content is extracted from lattice gluon configurations. Finally, both comparison of results including all gluons with those including only instantons and the study of quark zero modes associated with instantons and their contributions to hadronic observables are used to show the dominant role of gluons in hadron structure. MIT CTP# 2701 hep-lat/9804017 1. – Introduction and Motivation Although a quarter of a century has passed since the experimental discovery of quarks and the formulation of QCD, we are only now beginning to understand the essential physics of the structure of light hadrons. To truly understand hadron structure, one must solve rather than model QCD, and the only known means to do so is the numerical solution of lattice field theory. But obtaining accurate numerical results for observables from a computer is not enough — we also need to obtain physical insight. Hence, our strategy is to use numerical evaluation of the QCD path integral on a lattice to identify the configurations that dominate the action as well as to calculate observables. In recent years, the algorithms and techniques of lattice QCD and the performance of massively parallel computers have developed to the point that we are now on the threshold of reliable, quantitative calculations of QCD observables. Furthermore, there is strong evidence from lattice calculations that the topological excitations of the gluon field corresponding in the semiclassical limit to instantons play a dominant role in the structure of light hadrons. The purpose of these lectures is to describe at an elementary level the basic elements of lattice QCD and how numerical solution of QCD on a lattice is elucidating the role of instantons in light hadrons. To appreciate the significance of the current lattice results, it is useful to recall the wide range of disparate physical pictures that have arisen from the different QCD inspired models introduced to model hadrons. For example, non-relativistic quark models focus on constituent quarks interacting via an adiabatic potential. Bag models postulate a

A new Fourier path integral method, a more general scheme for extrapolation, and comparison of eight path integral methods for the quantum mechanical calculation of free energies

Using an isomorphism of Coalson, we transform five different discretized path integral (DPI) methods into Fourier path integral (FPI) schemes. This allows an even-handed comparison of these methods to the conventional and partially averaged FPI methods as well as a new FPI method. It also allows us to apply to DPI methods a simple and highly effective perturbative correction scheme (previously presented for FPI methods) to account for the error due to retaining only a finite number of terms in the numerical evaluation of the propagator. We find that in all cases the perturbative corrections can be extrapolated to the convergence limit with high accuracy by using a correlated sequence of affordable calculations. The Monte Carlo sampling variances of all eight methods studied are very similar, but the variance of the perturbative corrections varies markedly with method. The efficiencies of the new FPI method (called rescaled fluctuation FPI) and one of Fourier analog methods compare favorably with that of the original FPI method. The rescaled fluctuation method not only proves practically successful, but it also gives insight into the origin of the dominant error in the conventional FPI scheme.

Eigenstates from the discretized path integral

A numerically convenient and accurate method for evaluating eigenstates of a quantum system with an arbitrary angular momentum ( l) is proposed using the discretized version of the path integral. The accuracy of the method is examined by comparing the calculated results with analytical solutions for simple systems.

An ab initio path integral Monte Carlo simulation method for molecules and clusters: Application to Li4 and Li5+

A novel method for simulating the statistical mechanics of molecular systems in which both nuclear and electronic degrees of freedom are treated quantum mechanically is presented. The scheme combines a path integral description of the nuclear variables with a first-principles adiabatic description of the electronic structure. The electronic problem is solved for the ground state within a density functional approach, with the electronic orbitals expanded in a localized (Gaussian) basis set. The discretized path integral is computed by a METROPOLIS Monte Carlo sampling technique on the normal modes of the isomorphic ring polymer. An effective short-time action correct to order τ4 is used. The validity and performance of the method are tested by studying two small lithium clusters, namely Li4 and Li5+. Structural and electronic properties computed within this fully quantum-mechanical scheme are presented and compared to those obtained within the classical nuclei approximation. Quantum delocalization effects turn out to be significant as shown by the fact that quantum simulation results at 50 K approximately correspond to those of classical simulations carried out at 150 K. The scaling factor depends, however, on the specific physical property, thus evidencing the different character of quantum and thermal correlations. Tunneling turns out to be irrelevant in the temperature range investigated (50–200 K).

Circumventing the pathological behavior of path-integral Monte Carlo for systems with Coulomb potentials

The imaginary-time discretized path integral for systems interacting through Coulomb potentials and/or bounded by hard walls behaves pathologically. In this paper we give an analysis of the pathologies. A new method is used to calculate the path integral for the Coulomb system within the primitive algorithm. This is done by constructing an effective potential that implicitly depends on Trotter number P. The procedure makes possible the treatment of Coulombic potentials by means of Monte Carlo at surprisingly low Trotter numbers and without the need of computing three-body potentials, necessary when high-order approximants for the density matrix are employed.

High-accuracy discrete path integral solutions for stochastic processes with noninvertible diffusion matrices

The derivation of the discrete path integral solution for the propagator is known to present a special problem for those stochastic processes whose diffusion matrices are noninvertible. In this paper two methods for formulating the stochastic dynamics in terms of path integrals are developed that are applicable whether or not the diffusion matrix is invertible. One of the methods is an extension of the standard technique available for the derivation of the functional formalism from Langevin equations. An accurate discretization scheme is used to replace these equations by finite-difference equations and a short time approximation for the propagator is then derived in terms of known statistical properties of noise terms. An alternative derivation of the discrete path integral is presented in terms of the Fokker-Planck formulation without the necessity of introducing discretization schemes into the discussion. This is achieved by making use of the cumulant generating function which is different in this realm. The mutual correspondence of the methods is established and their possible extensions are discussed. Both methods are indeed rigorous and allow for the systematic derivation of the short time propagator valid to any desired precision in a time increment \ensuremath{\tau}. Its use in a path integral means a significant reduction of the number of time steps that are required to achieve a given level of accuracy for a given net increment t=N\ensuremath{\tau}, and, therefore, significantly increasing the feasibility of path integral calculations. Another attractive feature of the present techniques is that they permit the efficient treatment of equations with singular diffusion matrices, two of which, a Kramers equation and a colored-noise problem, are considered.

Path Integral Solution of the Kramers Problem.

An iterative method to generate a discrete path integral solution of the Kramers problem is presented. It is based on a straightforward derivation of the functional formalism from the underlying Langevin equations. The method is rather simple and systematic and allows us to analytically evaluate the short time propagator up to and including terms of fourth order in a time increment $\ensuremath{\tau}$. This means a significant reduction of the number of time steps $N$ that are necessary to obtain convergent results for a given net increment $t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}N\ensuremath{\tau}$.